Why can we use a network to estimate $Q_\pi(s, a)$ in Actor-Critic Method?
The policy doesn't change over time. That is, the values will change, otherwise we would not be learning anything, but our rules for action selection don't. I.e. we always take action according to the distribution postulated to our current estimate of the policy $\pi_\theta(a|s)$, we don't suddenly start taking $\max_a \pi_\theta(a|s)$, which would be a true change in policy and would make learning both the actor and the critic unstable.
Because NN's are able to handle noisy target distributions, this is how they can deal with the changing data. If you think of how Actor-Critic methods work, you would initially start to shift your NN to some unfeasible values (due to random initialisation and the Actor-Critic not having any information about the environment), but as you start to interact with the environment you will start to update the agent towards the 'true' policy.
An analogy in supervised learning would be to have some noisy data which is incorrect and some true data. If you trained your network on the noisy data for a small number of epochs and then never showed it to the network again and trained it solely on the correct data, it would forget it has ever seen the noisy data and only represent the new, true data.